A NNALES SCIENTIFIQUES DE L ’É.N.S.
W ILLIAM C ASSELMAN F REYDOON S HAHIDI
On irreducibility of standard modules for generic representations
Annales scientifiques de l’É.N.S. 4e série, tome 31, no4 (1998), p. 561-589
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46 serie, t. 31, 1998, p. 561 a 589.
ON IRREDUCIBILITY OF STANDARD MODULES FOR GENERIC REPRESENTATIONS
BY WILLIAM CASSELMAN AND FREYDOON SHAHIDI (*)
ABSTRACT. - In this paper the authors generalize a result of Vogan on irreducibility of standard modules for generic representations from real groups to p-adic ones whenever the inducing data is supercuspidal. They also prove injectivity for standard modules in this case. As applications, the authors prove a number of results relating the poles of intertwining operators and points of reducibility of induced representations to the poles of L-functions denned by the second author, modulo a conjecture on them whose validity for classical groups is also verified here. A result on certain real groups with applications in liftings of automorphic forms from classical groups to general linear groups via L-functions is also proved. © Elsevier, Paris
RESUME. - Dans cet article les auteurs generalisent un resultat de Vogan sur Firreductibilite des modules standards pour les representations generiques des groupes reels vers les groupes j?-adiques quand 1'induite est supercuspidale.
Us prouvent egalement Finjectivite pour les modules standards dans ce cas. Les auteurs en deduisent quelques resultats reliant les poles des operateurs d'entrelacement et les points d'irreductibilite des representations induites, aux poles des fonctions L definies par Ie deuxieme auteur, modulo la validite d'une conjecture que 1'on verifie ici pour les groupes classiques. On montre aussi un resultat sur certains groupes reels avec applications aux correspondances des formes automorphes des groupes classiques vers les groupes lineaires via les fonctions L.
© Elsevier, Paris
Introduction
The purpose of this paper is to prove a number of results in representation theory and harmonic analysis of local groups, some of which have important consequences in the theory of automorphic forms.
More precisely, let G be a quasisplit connected reductive group over a local field F of characteristic zero (real, complex, or p-adic) and let B = TU be a Borel subgroup of G, where T is a maximal torus of B and U is its unipotent radical. Let Ao be the maximal split torus of T. Fix a parabolic subgroup P of G defined over F with a Levi decomposition P = MN, with T C M and N c U. Let a be an irreducible tempered representation of M = M(.F) and choose v G a^, the complex dual of the real Lie algebra of the split component A of M. (See Section 1). Let J(^a) be the representation (unitarily) induced from v and cr. Assume v is in the positive Weyl chamber (Section 1). Then J(^ a) is called a standard module. Let J(z/, a) be the (unique) Langlands quotient of I { y , a) (cf. [4, 17, 28]). Up to conjugation of the data (y, a), every irreducible
(*) Partially supported by NSF Grant DMS-9622585.
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admissible representation of G = G(F) is uniquely equivalent to a J(^,a). Moreover, every irreducible admissible generic (having a Whittaker model) representation of G is a
J ( y ^ a) with a an irreducible generic tempered representation of M.
When F = R, it was proved by Vogan in [37] that if e7(^,a) is generic, then J(^a) = J(^cr), i.e. I(^,a) is irreducible. The aim of this paper is to address this question and related ones for p-adic groups. In fact, in this paper we prove (Theorems 2.2 and 3.4):
THEOREM 1. - Suppose a is an irreducible unitary generic supercuspidal representation of M and fix v € d^ in the positive Weyl chamber.
a) Assume J(^,cr) is generic. Then J(^,cr) == J(^,a), i.e. J(^,cr) is irreducible.
b) Suppose J{y^ a) is not generic. Then the unique irreducible subspace of J(z^ a) is generic.
Equivalently, irreducible generic subquotients of I{v^ a) are subrepresentations and are therefore unique.
In the special case that P = B, i.e. is a minimal parabolic subgroup and a is an unramified quasicharacter of M = T, Part a) of Theorem 1 was proved in [2], [20], and [22], earlier, each using a different method. We refer to [3], [14], and [41] for G = GLn, but general tempered a.
When G = Sp^n or 5'02n+i. part a) was recently proved in the generality of arbitrary standard modules by Goran Muic in a very interesting manuscript [42]. As is the case with our results, his proof is based on the theory of L-functions developed in [23].
Although the result stated above is only for inducing supercuspidal data, the theorem is expected to be true in the generality of every standard module if part b) is formulated as:
1) Irreducible generic constituents of standard modules are subrepresentations. (See the remark after Definition 3.1.)
We have called this statement the generalized injectivity conjecture (Conjecture 3.3) and it is clear that it implies part a), i.e.
2) Standard modules for generic representations are irreducible.
Theorem 3.4 then proves this for inducing supercuspidal representations.
To state some applications of injectivity (e.g. Theorem 5.1), one needs to discuss a conjecture (Conjecture 7.1 of [23]) whose validity also plays an important role in the proofs given here and [42].
To explain the conjecture, assume P is maximal. But a is any irreducible admissible generic representation of M. If r is the adjoint action of LM, the L-group of M on the Lie algebra Ln of the L-group of N, then r = (D^i^, with r^s ordered as in [23], i.e. according to the order of eigenvalues of LA in Ln. Finally, for each %, 1 < i < m, let L(s^a^ri) be the local L-function attached to a and ri as in [23]. When F = R, the L-functions are those of Artin (cf. [1,17,18,24]). (See Section 6 here and Theorem 3.5 and Section 7 of [23].) Conjecture 1.1 then demands that each L(s^a^ri) be holomorphic for Re(s) > 0 whenever a is tempered.The normalized intertwining operators then satisfy the last condition Rj set forth by Arthur in [I], whenever normalization is as in [19,23].
This conjecture has been verified in many cases in [23], including when m = 1 or a is supercuspidal, and the subject matter of Section 4 of the present paper is to prove it whenever G is of classical type. This includes all the quasisplit classical groups.
^ SERIE - TOME 31 - 1998 - N° 4
Next, let J(^,cr) be a standard module attached to a tempered representation a- of M, where P = MN is the corresponding standard parabolic subgroup. Denote by WQ the longest element in the Weyl group of Ao in G modulo that of Ao in M. Fix a representative wo for WQ in G. Let A(^,cr,wo) be the standard intertwining operator from J(^,cr) into J(wo(^),wo(cr)) (cf. Section 1). It is well defined since v is in the positive Weyl chamber.
But its continuation to all of a^ may have poles and that is where the problem lies. Assume P is maximal. Let a be the unique simple root in N. Set p = {p, a)~1?, where p is half the sum of roots in N. Fix s G C. We may take v = sa € a^. Then Re{s) > 0 since v is in the positive Weyl chamber. Consider the operator
m
(3) ^] L(is, a, r^A^sa, a, Wo) 1=1
on I(sa^a).
The homomorphy of (3) for all s G C has important global consequences through normalization of intertwining operators and Eisenstein series (cf. [II], [15], [19], [21], [30], [31], [43]), and although at present we are unable to prove it in general, there are practical instances when this can be accomplished. In fact, our Theorem 5.1 proves the holomorphy of (3) on all of C under what we call injectivity (Definition 3.1) for all the corresponding rank one standard modules. (See the remark after Definition 3.1.)
One important instance when Theorem 5.1 can be applied is when F == R or C, and G = 502n, the split even special orthogonal group of rank n (Theorem 6.1). The case in hand has an important application in the project of lifting automorphic forms from classical groups to general linear groups as being pursued in [11, 30, 31], using the converse theorem for L-functions [9].
To check that the hypothesis of Theorem 5.1 is satisfied, it is sufficient to prove that standard modules for GLnW satisfy injectivity, i.e. their irreducible subrepresentations, which turn out to be a single one, are all generic (large). A proof of this was communicated to us by Vogan. We would like to thank him for providing us with a proof and allowing us to include it here.
Vogan's proof is quite instructive. It relies on cohomological induction and is therefore algebraic. On the other hand, after communications with him, the authors realized that there is an analytic proof of Theorem 6.2 due to Jacquet and Shalika (Proposition 4.2 of [13]) which relies on the theory of canonical extensions of Harish-Chandra modules as developed by Casselman [6] and Wallach [40]. The existence of this second proof, which in spirit is closer to our approach in Section 3, was envisioned and communicated by the first author to several people, many years ago (cf. [7], for example).
Our final application, Proposition 5.3 (and 5.4), determines the points of reducibility for every I(sa, a) in terms of poles of L-functions, but under the assumption of validity of (2).
When F = R, the assumption is already proved, and therefore Proposition 5.3 shows that
m
points of reducibility for I(sa,o-), Re(s) > 0, are precisely poles of ]~[ £(1 - %5,a,r,), where Z-functions are those ofArtin, attached to a and r, by Langlands [T], [17], [18], [24].
Proof of Conjecture 1.1 for the classical groups, given in Section 4 here, relies to certain extent, on what cuspidal inducing data for discrete series are. Except for Lemmas 4.1 and 4.6 of [34], we have relied entirely on our own method to determine them and our results are quite parallel to those of Tadic [34], [35], [36]. The relation between our
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
two methods must certainly be explored further. (See Remarks 4.21, 4.22, and 4.23 here and [42].) Lemma 4.1 has also been verified independently by Y. Zhang in a work in progress, using results of Harish-Chandra and Silberger on special orbits (cf. [28]).
Finally, since the ^-functions are supposed to remain the same for members of each tempered L-packet (Conjecture 9.4 of [23]), one must explore the possibility of extending such results to non-generic representations. The project which has been started in [10]
and is aimed at developing similar results for non-generic representations may eventually provide us with some evidence in this direction.
The second author would like to thank Joachim Schwermer for his hospitality during his visit to Eichstatt in the summer of 1993. In fact, his renewed interest on the problem was fired up by questions posed to him by Schwermer.
We would like to thank David Collingwood, Birgit Speh, Marko Tadic, and David Vogan for several useful communications. We would also like to thank Jean-Loup Waldspurger for communicating to us Tadic'& original counterexamples to our original version of injectivity.
1. Notation and Preliminaries
Let F be a local field of characteristic zero. When F is nonarchimedean, we use q to denote the number of elements in its residue field.
Throughout this paper, with the exception of Section 4, G denotes an arbitrary quasisplit connected reductive algebraic group over F. Fix a Borel subgroup B and write B = TU, where T is a maximal torus and U denotes the unipotent radical of B.
Fix a parabolic subgroup P == MN of G defined over F with N C U and T C M, a Levi decomposition. Let Ao be the maximal F-split torus of T and denote by W(Ao) the Weyl group of Ao in G. Let wo be the longest element in W(Ao) modulo that of the Weyl group of Ao in M. Let '0 be a generic character of U = U(F) (cf. [8,23]) and set ^M = ^\U D M. Suppose a is an irreducible admissible ^M-generic representation of M = M(F). Changing the splitting in U we may assume that ^ and WQ are compatible (cf. [23]).
Let X(M)p be the group of F-rational characters of M. Set a* = X(M)F 0z
and
a^ = a* (g)R C.
As usual (cf. [23]), we let
I{y, a) = IndMTVfG^ ^ ^p( )> 0 1,
where v G a^ with exp replacing q if F = R. Here Hp is the extension of the homomorphism
HM : M -> a = Hom(X(M)F, R) to P, extended trivially along N, where HM is defined by
^P(-)) ^ |^)|^
for all x ^ X(M)F.
46 SERIE - TOME 31 - 1998 - N° 4
Suppose P = MN is maximal, M D T. Let r be the adjoint action of LM, the L-group of M, on the Lie algebra Ln of the L-group of N. Then r = (]) r,, with r,'s irreducible,
1=1
ordered as in [23], i.e. according to the order of eigenvalues of LA in Ln. Here ^ is the L-group of A, the split component of M. It is contained in Ao. Finally, let L(5,a,r,) be the local L-function attached to a and r, in [23]. (See Theorem 3.5 and Section 7 of [23].) Conjecture 7.1 of [23] plays a crucial role in the present paper and therefore for the convenience of the reader we state it here once more.
Conjecture 1.1. Assume a is tempered and Re(s) > 0. Then each L(s,a,ri) is holomorphic, 1 < i < m.
Next, assume M is generated by a subset 6 of simple roots A of Ao in U. Fix w G W{Ao) ^uch that w((9)_(: A and let w e G be a representative for w. Let N^ = U D wNw~1, where N is unipotent subgroup opposed to N. Given / in the space of J(^a), let
A{^,a,w)f(g)= f(w~lng)dn (g G (?) J N -
denote the standard intertwining operator from I{y,a) into J(w(^),w(cr)). It converges absolutely in some cone and extends to a meromorphic function of v G d^ (cf. [12, 16, 29]). The knowledge of its poles on all of a^ is very important and one of the aims of the paper is to determine them in terms of L-functions mentioned before in certain cases.
When a is tempered the cone of convergence for v G a^ equals to what one usually calls the positive Weyl chamber (a^ for a. Every v e (a^)+ satisfies Re(v,Ha} > 0 for every a G A - 0 and conversely, where H^ is the standard coroot attached to a and u is realized as an element of (do),^. Here do is the real Lie algebra of Ao.
Suppose a is tempered and v e (a^. Then J(^,cr) has a unique quotient J(^cr), called the Langlands quotient of J(^,a) (cf. [4, 17, 28]). Given an irreducible admissible representation TT of G, there exists a parabolic subgroup P = MN, N c U, M D T, an irreducible tempered representation a of M, and a v G (d^)+, such that TT = J(i^,a).
Moreover, by Rodier's Theorem, TT is generic only if a is.
Since a part of this paper is heavily based on material in [5] and [8], we will adopt their notation in the following definitions for the convenience of the reader.
Let A be the set of simple roots for M^ = Ao in U = N^. Fix a subset 0 C A and let P =Pe = M^N^ = MN, N^ c N^,, be the corresponding standard parabolic subgroup.
Let ^ be a nondegenerate character of A^>, extending a nondegenerate character ^ = ^Me of Me n N^. Let WQ be a representative for the element WQ of the Weyl group of Ao which sends every root in A - 0 to a negative one, while wo(a) e A for all a e 0. Let M' = M^^ and denote by P' = M'N', N' c N^,, the standard parabolic subgroup of G having M' as its Levi factor. Assume ^ and WQ are compatible (cf. [23]). This can always be achieved by changing the splitting in N(^.
Let (a,^(cr)) be an irreducible admissible ^-generic representation of M = Me. Fix a Whittaker functional OM for a. Let P be the parabolic subgroup of G opposed to P.
Then P = MN, N = N_^. One can then define standard Whittaker functionals ^ and H on I{a) = Ind(a|P,G) and I(a) = Ind(a|P,G), respectively by
(i-i) (f^}= I ^wa^o"
J N '1 ^), "M^
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f G J(a), as in [8, 25], and
(1.2) (7,») = / ^-\n){f{n)^ ^dn^
J N
when / G ^(cr). Both integrals are defined as principal values (cf. [8]).
Let
T:7(a)^J(a) be the intertwining operator
(1.3) Tj{g) = ( ~f{ng)dn J N
determined by the 7-((a)-valued functional
AN (7)- l~fWn.
Then T/(e) = A^(/). Let 7 be the constant defined by Rodier's theorem, i.e.
(1.4) T*^ = 7^.
Next let a' = wo(a) and let C^{(T') = C^(a',Wo"1) be the local coefficient attached to a', Wo'1, and '0 (cf. [25]). More precisely it is defined by
{f^ff rV\ _ r^ ( ^'\l \( ^1 ...-1\ fif)=C^af){A(a\w,)f^ n), i ^ ^ l i
where A(cr',Wo"1) : Ind(a/ P^G) —> Ind((T|P,G?) is the standard intertwining operator and W is the standard Whittaker functional on J(a') = Ind^'l?', G). Finally, let 7 = 7(0) be as in equation (1.4). We have:
LEMMA 1.2. - 7 = C^^Wo'1)"1. Proof. - Let 7(cr) = Ind(a|P,G) and let
T : 7 ( a ) ^ J ( a ) be the intertwining operator (2.3). If
L^: J(a')^7(a) is defined by L^f{g) = f(wog), then
(1.2.1) r.L^^A^Wo-1).
By definition
(1.2.2) <7,T*n}=7<7,"), where 7 = ^wo/', // ^ ^(^/). Then
(1.2.3) {L^ff^}={ff^f)
4e SfiRIE - TOME 31 - 1998 - N° 4
with 0' as above. Using (1.2.1) and (1.2.3), (1.2.2) implies {A{a^w,l)ff^)=^f^f)^
from which the lemma is immediate.
If a is a ^-generic irreducible tempered representation of M and v G a^, we use C^(^a) to simply denote the local coefficient [25] attached to V,(T, and wo, i.e. the complex function defined by
(f^)=C^a){A^^wo)f^f}, where
A(^cr,Wo) : I{^a) -^ J(wo(^),Wo(cr)) is the standard intertwining operator defined earlier.
2. Irreducibility of standard modules
Let TT be an irreducible admissible generic representation of G. By Langlands classification ([4, 17, 28]) there exists a standard parabolic subgroup P = MN of G with split component A, an irreducible generic tempered representation a of M and a v in the positive Weyl chamber of a^ such that TT = J(^,a), the (unique) Langlands quotient of J(^cr). When F = R, in [37], Vogan proved:
THEOREM 2.1 (Vogan). - Assume F = R and TT = J(^cr) is generic. Then J{v,cr) = J(^, a). In particular I(y^ o) is irreducible.
Now suppose F is p-adic. In this section we extend Theorem 2.1 to p-adic fields when a is supercuspidal. More precisely, we prove:
THEOREM 2.2. — Let a be an irreducible generic unitary supercuspidal representation of M. Assume J(^,cr) is generic. Then I{y^cr) =• J(^,a), i.e., I(v^(r) is irreducible.
Proof. - Assume a is ^-generic and let ^ be an extension of z^e (cf. Section 1).
Let C^(wo(^),wo(cr)) be the corresponding local coefficient, i.e. the complex number defined by
(2.2.1) (//, n') = ^(woM, wo(a))(A(woM, wo(a), Wo-1)/', 0),
where f^' is the standard Whittaker functional on J(wo(^),wo(cr)). By Proposition 7.3 of [23], Conjecture 1.1 is true and therefore the denominator of C^(wo(^),wo(<7)) is a product of the form
]"j£(l+^wo(cr),^-)~\
3
where ^-functions are as in [23] with Re (sj) > 0, since v is in the positive Weyl chamber (Theorem 3.5 of [23], equation (3.11)). Again by Proposition 7.3 of [23], this product is non-zero since these L-functions are holomorphic if Re(sj) > 0 (in fact if Re{sj) -^- —1,
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a- being supercuspidal). In conclusion C^(wo(^),wo(cr)) is well defined whenever v is in the positive Weyl chamber.
On the other hand, by Theorem 5.4.2.1 of [29], the operator A(wo(^),wo(a),Wo~1) is holomorphic as long as a is unitary supercuspidal and v is regular, and in particular if v is in the positive Weyl chamber. It now follows that the normalized operator
(2-2-2) Cv,(wo(^), wo(a))A(wo(^), wo(a), WQ1)
is well defined on all of J(wo(^),wo(cr)) if I(i^,a) is standard.
Choose // in the space of J(wo(^),wo(cr)) such that (f,^) / 0. Then by equation (2.2.1), the image of /' under (2.2.2) is ^-generic.
Suppose J(^a) is ^-generic, but J(^cr) is reducible. The image of I(wo(i/),wo(a)) under (2.2.2) being ^-generic, will have an irreducible ^-generic subquotient which is inequivalent to J(^cr) by uniqueness of Langlands quotient. This contradicts Rodier's Theorem and proves the Theorem.
Remark 2.3. - The standard modules which are built by means of minimal parabolic subgroups are clearly among special cases covered by Theorem 2.2, since quasicharacters of A<^, the F-points of the split component of M^ are supercuspidal. When a is an unramified character of A<^, the theorem was proved in [2], [20], and [22], earlier, each using a very different method.
COROLLARY 2.4. - Let 7 be as in equation (1.4) and assume I(a) = lnd(a\P^G) is standard. Then 7 / 0 .
Proof. - This follows from Lemma 1.2 and the fact that C^ (a') = C^(wo(a)) is well defined which was observed in the proof of Theorem 2.2, if a is in the positive Weyl chamber.
3. Generalized injectivity
In this section we will address a property of standard modules which implies Vogan's theorem for them and is therefore rather stronger. It simply requires the generic constituents of a standard module to become subrepresentations.
DEFINITION 3.1. - A standard module is said to satisfy injectivity if all its irreducible subrepresentations are generic.
Remark. - When the inducing data is supercuspidal, we will show (Theorem 3.4) that the corresponding standard module satisfies injectivity. We initially believed that every standard module for a p-adic group satisfies injectivity. In fact, as opposed to the case of real groups for which already the rank one quasisplit group (7(2,1) has standard modules which do not satisfy injectivity [4], standard modules defined by supercuspidal data for any p-adic quasisplit group do. But certain recent examples of Tadic for GSp^n^ n >_ 4, has convinced us to contrary in general. In fact, Tadic has a class of counterexamples for classical groups which are now included in his new version of [34]. What follows are some low rank examples which were communicated to us by him.
The lowest rank examples of Tadic are for GSps{F) and SO^^F). In both groups T = AQ acts transitively on the set of generic characters of U. Let a-o denote the Steinberg
46 SERIE - TOME 31 - 1998 - N° 4
representation of GLn(F)^ n = 3 or 4, a non-supercuspidal discrete series representation. If G = GSps(F),\eta = ao0|det() |^ 01, a representation of GL^F)xGL^(F), the Siegel Levi subgroup of GSps(F). On the other hand for G = SO^{F\ let a- = o-o 0 det( )|^
denote one for GLs(F), the Siegel Levi subgroup of SOj(F). In either case, using Jacquet modules, Tadic shows that the standard module J(a) has two (non-isomorphic) irreducible subrepresentations. By the above remark one of them must be degenerate.
We therefore need the following definition.
DEFINITION 3.2. - A standard module is said to satisfy generalised injectivity if all its generic constituents appear as subrepresentations.
CONJECTURE 3.3. - Every standard module satisfies generalised injectivity.
Remark. - The conjecture seems to be valid for real groups although no proof for it has been published (private communications with Vogan). In this section we will prove the conjecture when F is p-adic and the inducing data is supercuspidal.
THEOREM 3.4. - Let F be a p-adic field of characteristic zero and let G be a quasisplit group over F. Let I{y^ a) be a standard module for G. Assume a is generic and supercuspidal. Then the unique irreducible subrepresentation of I(i^,a) is generic, i.e., injectivity is valid for supercuspidal inducing data. In other words J(z^, a) injects into Indu^G^ for every ip extending ^e, the character with respect to which a is generic.
We use notation as in Section 1. More precisely, let
^ =a^q^HM{ ) >.
Then we set J(cr^) = I(a^) and use I and V to denote I(a^) and the space V(a^) = V(^a) of I(^a).
Since a^ is supercuspidal and regular. Proposition 6.4.1 of [5] implies that a^61/2
appears in VN = V^e, the Jacquet module of V with respect to N , with multiplicity one.
Consequently by Frobenius reciprocity V has a unique irreducible subspace. Theorem 3.4 then requires this subrepresentation to be generic and in particular it will be generic with respect to every extension ^ of ^0. We start with two lemmas. Again the notation is as in Section 1.
LEMMA 3.5. - Suppose f e Ind(cr|P, G) has support in PN. Then there exists e = e(f) > 0 such that
(I(a)~f^) = 61p\a)a^a)(AN(7)^ OM)
for every a G Ae with \a(a)\ < £, a e A — 0. Here Ao is the split component of M@, I = I{o'v), and 6p is the modulus character of P.
Proof. - Assume / is supported in PUJ. Then {7(a)7,n)= / ^{^(/(na)^ ^dn
J N
= f ^(^^^(^(^(/(a-^a), ^M}dn J N
= f ^(ana-^^Wa^a)^^), ^dn.
J N
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Now choose e small enough so that ^|ao;a~1 = 1, concluding the lemma.
Let V be the space of Z(cr^) and denote by V^o its Jacquet module with respect to N = Ne, where 0 is the subset of A generating M = Me. The space of Whittaker functionals on V^e is the dual of (V^J^^Men^ (notation as in [8]).
With notation as in [5] and [8], consider the diagram
V V^)/V^Ne)
^ ^ ^
^Ne ———^ ^0,M?
P\ \ P^
^ ^6 '
(^Ae)^,MonA^ ——^ ^,A^>
1
in which (f)e is defined by means of canonical liftings (cf. Section 4 of?]), exactly as in the minimal case in [8]. It can be easily shown that ker (/?) C ker (p^ • <^), leading to the well defined map (f)e of Proposition 6.4 of [8j. Consequently 0 • (f)e defines a functional on (y/v^v^MonA^ and is therefore a Whittaker functional for V N ( ) ' Here 0 = A^(^,a) is the canonical Whittaker functional for I ( y ^ < J ) defined in Section 1. We now restate Proposition 6.4 of [8] in our case as follows.
LEMMA 3.6. - Given f in the space V of I(v^ a), there exists e > 0 such that (I{a)f^)=^e{lNeWf)
for all a € Ae with |a(a)| < £, Va G A — 0. Here f is the image off in V^g.
Proof of Theorem 3.4. - By Proposition 6.4.1 of [5]
(3.4.1) I N = I N C = © (wa,)51/2,
wCH^(A)
where the sum is over the Weyl group of A in G. Since a^ is regular, each irreducible subrepresentation appears with multiplicity one in (3.4.1). Given / 6 I{o~^), write
/- ©^
w
where fw is in {wa^)S1/2. Here f is the image of / in J^v. By Frobenius reciprocity W^)7i = ^{m}6^\m)f{e} (m 6 Me}.
For each w, let \w be a Whittaker functional for (wa^)^1/2. We will take Ai = ^IM' Write
^ • ^e = y C w A w ,
4e SERIE - TOME 31 - 1998 - N° 4
using the fact that 0 • ^ is a Whittaker functional for VN^. By Lemma 3.6 there exists e = e{f) > 0 such that
(3.4.2) {I{a)f^) = n.^(J^(a)/) = ^^(wa)(a)51/2(a)<7„A,)
w
for all a € Ae = A with |a(a)| < e, Va G A - 0.
Next, choose / € ^(^) with support in PN such that T/(e) / 0. Such functions exist by Lemma 4.2 of [27]. By Lemma 3.5 and equation (1.4), there exists e = e{f) > 0 such that
(3.4.3) <^)(m ") = 7^1/2(a)<(^7)(e), ^)
for a ^ Ae with ^(a)! < e, Va G A - 6.
Since a^ is regular. Theorem 5.4.2.1 of [29] implies that T ' L^ is well defined on all of J(wo(^),wo(cr)) where
T : 7(a,) -^ J(a,) is defined by (1.3) and
^wo ^ -f(<^) = ^(wo(^),wo(cr)) ^ 7(a^)
is as in Lemma 1.2. Consequently / = Tf is well defined and belongs to J(cr^). Comparing (3.4.3) with (3.4.2) and using the regularity of a^, one has ci = 7 which is non-zero by Corollary 2.4. We can therefore write
(3.4.4) <J(a)/, n) = 7a^l/2(a)</(e), ^) + ^ ^ . (wa^a^/^aX/w, A,).
w^l
Since v is in the positive Weyl chamber, the term 7^1/2(^(/(e)^M)
is now a leading term as |a(a)| —^ 0, a € A^, Va G A - 0, if (/(e), ^M) 7^ 0-
Suppose 0 / / G V is such that {I(g)f,a} = 0 for all g € G. Then IVj = 0, where Wf{g) = (I{g)f,fl), i.e., 0 / / lies in the kernel of f \—^ Wf into Ind^G'0 or injectivity fails.
We may assume </(e), ^) 7^ 0. Then for a e Ae with |a(a)| < e = £(/), Va G A - 0,
0 = <J(a)/, ^) = 7^^1/2(^</(^, "M) + • • •
with ^o•^61/2(a)(f{e)^M), a non-zero leading term as |a(a)| H-^ 0, Va G A — 0. This is a contradiction.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUP^RIEURE
4. Proof of Conjecture 1.1 for Groups of Classical Type
Throughout this paper, a classical group is a connected algebraic group, fixing a non- degenerate bilinear form of either symmetric, alternating, or Hermitian type. The group G will be called of classical type if there exists a product of classical groups whose derived group is a covering, as an algebraic group, of the derived group of G. We will further assume that G is quasisplit. The purpose of this section is to prove Conjecture 1.1 when G is of classical type.
Let G be a quasi-split classical group over a p-adic field F of characteristic zero. Let a^ be a discrete series representation of G = G(F). Choose a Levi subgroup
M = GLr, x . . . x GL^ x G°
of G, with G° classical. When G is the unitary group defined by a quadratic extension E/F,GLr, must be replaced by Res^GL^. The reader must be warned that a^ is a pure symbol and is neither the dual nor the contragredient of a representation 02. Choose an irreducible supercuspidal representation 02 = pi (g)... 0/^ 0r of M = M(F) such that
a^ C Ind^a2 0 1.
Next, let ao be an irreducible unitary supercuspidal representation of GLt(F). Assume r is generic. Let a^ be a discrete series representation of GLu(F) with t\u, defined by means of (TO as in [3, 41]. (See below.)
We are interested in the Rankin-Selberg product factors for a\ x a^. More precisely we want to study
L(s,a^ x ^ ) ( 5 0 C)
as defined by the 7-function 7(5, a^ x o-^^p) of Theorem 3.5 of [23]. (See Section 7.) Here 7(5, a\ x a^^p) is 7(5, a\ 0 a^ r^ ^) of cases B^ C^ Dn - 1, ^ - 4, and
^Dn - 1, chosen according to G. For the sake of simplicity, from now on, we drop the dependence on ^p from 7(5,0^ x a^^p). As in Corollary 5.6 of [26] we have:
n
(4.1) 7(5, ao x (T^) = 7(5, ao x r) ]"J 7(5, (TO x pj)^{s, (TO x p j ) j=i
for the pair (o-o,^).
For each j, 1 < j < n, choose an irreducible unitary supercuspidal representation poj and a real number ^ such that pj = poj 0 |det( )|^:
Let p = pi 0 ... 0 pn and set
n
(4.2) 77(5, ao, r, p) = Z<5, ao x r) JJ L(s + ^, ao x poj)L(s - ^,0-0 x poj).
j=i
The reader must realize that when we are in the unitary case An - 4 of [23], po j must be replaced by poj throughout, where poj denotes the Galois conjugate of poj under the non-trivial element of the Galois group of the defining quadratic extension.
4^^ SERIE - TOME 31 - 1998 - N° 4
If A{s) and B(s) are two rational functions in q~8, we use A(s) ~ B{s) if they are equal up to a monomial in q~8. Then from (4.1) and (4.2) we have
(4.3) 7(5, o-o x a^) ~ 77(5 - 1, ao, r, p ) / r ] ( s , (TO, r, p).
Now suppose a^ is the unique discrete series constituent of
^^L^.N^l^--^^)^^
where nG£i(F) has b factors with 6 = u/t and cr, = o-o 0 | det( )|(&+1)/2-^ l < z ^ &.
(See [3, 41].) Then
b
7(5, ^ x ^v) = n 7(^ + (b + 1)/2 - z, ao x ^v) 1=1
b
= J J 7(5,0, X CT^) i=l
b
~ J]y?(5 + {b + 1)/2 - z - l,ao,T,/))/^(5 + (& + 1)/2 - z,<7o,T,p).
1=1
Define £(5,cr^ x a^)~1 as the numerator of 7(5, a^ x a^) as in [23]. We shall prove:
THEOREM 4.1. - The L-function L(s,a^ x a^) is holomorphic for Re(5) > 0, i.e. Conjecture 1.1 is valid.
The following corollary is then a consequence of part 2 of Theorem 3.5 of [23] and Theorem 4.1 here since the local coefficients for groups of classical type are just a product of those for classical groups. Observe that it states a result on the holomorphy of local coefficients which is usually deep.
COROLLARY 4.2. - Let G be of classical type. Then the corresponding local coefficient for any parabolic subgroup and any generic tempered representation of its Levi factor is holomorphic for v G '^o(((x^)+)•
By Lemma 4.1 of [34], representations poj are all self-contragredient. In fact, although in [34] this is only proved for symplectic and odd special orthogonal groups, we have been assured by Tadic that similar results can be proved for other classical groups. Similar remark applies to Lemma 4.6 of [34]. (See Remark 4.21 below.) Lemma 4.1 has also been verified independently in a work in progress by Y. Zhang, using results of Harish-Chandra and Silberger on special orbits (cf. [28]).
The L-functions in the product
n
JJ L{s + VJ,(JQ x poj)L(s - V j , ao x poj)
.7=1
are non-trivial only if (TO 0 |det( )\so^ ^ poj for some SQJ C %R. We therefore may assume, by shifting 5 by 5oj 6 %R, that (TO is self-dual and poj ^ ao. We shall therefore need to study products of type
n
L(5, (TO x r) TT L(5 + V j , (TO x ao)L{s - V j , (TO x 0-0), j-.=i
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
where ao ^ (TO and ^ G R. Observe that we may need to study these products with different values of s. But Re(s) will be the same for all such products.
Given t G C, set
L(t) = L{t, ao x ao) whose dependence on (TO is understood.
DEFINITION 4.3. - By a chain based on a-o or a cro-chain (simply a chain if 0-0 is fixed), we mean a sequence of representations
aj = a o 0 | d e t ( )p (^- e R )
for which vj - ^-i = 1 for all possible j. If (TO is understood to be fixed, we use {^}^=i to denote a chain based on ao. This is what is called a segment in [34] and [41].
Using results from [23], we shall first prove that one only needs a union of certain special types of chains to obtain all the induced representations which have discrete series subrepresentation. Our main tool is the following theorem whose proof is an application of Proposition 7.2.b of [23],, One observes that by Proposition 7.3 of [23] the assumption on L{s^(T,r^) = L(s^ao^r^) is in fact satisfied.
THEOREM 4.4. — Define L^s^ (TO x a^) as the inverse of the numerator 0/7(5, (TO x a^), where a^ is in the discrete series and o-o is irreducible unitary supercuspidal. Then
L{s,(To x a^) is holomorphic for Re(^) > 0.
We start with the following lemma.
LEMMA 4.5. - Fix complex numbers p, and v and let a^ = o-o 0 |det( )|^ and a^ = (TO ^ | det( )\^. Then L(s, (TO x (T^)~1 and L(s, (TO x cr^)"1 have a factor in common as polynomials in q~8 if and only if a^, ^ (Ty. In this case L(s^ao x (T^) = L^s^ao x (Ty).
Proof, - The L-function L{s,ao x ao) = n(1 ~ 7?(^7)(?-s)~l» where the product is over
r]
the group of all the unramified characters which fix (TQ. Then L{s, (TO x (T^) = L(s + ^, ao x o-o)
-n^-w ' r ] 0 ^)^)" 1 -
where ^ 0 = 1 1 ^ while
L(S,(TO x (T^ = 1[[{1 - rfrio^q-8
^
where rjo = | |^. If they have any factors in common, then rjo/rfo w1^ ^ave to belong to the group of unramified stabilizers of a and conversely.
Let {^j}^=i be a chain based on a self-dual irreducible supercuspidal representation o-o of GLt{F). Let a^ be a discrete series representation of G. Assume there exists a Levi subgroup M of G which is a direct product of an n-product of GLf with a Levi subgroup M' of a smaller rank similar classical group and an irreducible supercuspidal representation a^ of M' = W^F) such that a^ can be embedded as a subrepresentation of
4s SfiRIE - TOME 31 - 1998 - N° 4
the representation of G induced from (n^=i ^o ^ | det( )|^) 0 a^ of ]"[ GLt(F) x M1. We then say a^ is partially supported by the chain {^}7^.
DEFINITION 4.6. - Let {^-}^i be a ao-chain. We shall call
7^1 r } - TT L{s ~ uj ~ ws + yj ~ 1) 7<°5 ^l? • • • i ^n) — I | ~T~(————r-,-7—;——^———
,=l ^ - ^W + ^)
^ L(s -i^n- l)L{s + ^i - 1) L ( 5 - ^ i ) £ ( 5 + ^ ) the 7-function of the chain { ^ j } ^ i .
DEFINITION 4.7. - Two ao-chains are called 7-equivalent, if they have equal ^-functions.
LEMMA 4.8. - Every non-positive chain is ^/-equivalent to a non-negative one.
Proof. - Use 7(5; ^ i , . . . , u^) = 7(^; -^n, • . . , -^i).
LEMMA 4.9. - If there exists only one chain {^}^=i with z/i >_ 0 g^'vm^ ^ ao-support of a^, then either v\ = 0, ^i = 1/2, or ^i = 1.
Proof. - If z/i > 0, then by Theorem 4.4, L{s - ^i) must not appear in 7(5, o-o x a^).
By Lemma 4.5 we may assume that, either L{s — ^i)~1 cancels L{s + v\ — 1)~1, or it divides L{s - I,(TQ x r)~1. In the first case, again by Lemma 4.5, we may assume z^i - 1 = -^i, since their difference fixes o-o. The other case could only possibly happen if z^i = 1. One needs only to observe that L{s,ao x r) has no poles for s G (-1,0), a consequence of Proposition 7.3 of [23].
DEFINITION 4.10. - Fix o-o. Let {^}^=i be a non-negative chain in the ao-support of a^.
We shall say {^j}^=i is regular if either ^i = 1/2 or v^ = 0, or z/i = 1 and L{s)~1 divides L{s,ao x r)~1 (Lemma 4.9). Observe that by [23] this last condition (on L-functions) implies that the representation induced from (o-o 0 |det( )|) 0 T, i.e. va^ xi T in the notation of [34], is reducible. (See Theorem 3.3 of [34].)
DEFINITION 4.11. - Fix (TQ. Let {^j}^i and {^k}T=i be a pair of non-negative chains in the o-o-support of a^. Assume neither ^i = 1/2 nor v[ = 1/2. We then call {^} and {z^} a pair of singular chains, or a singular pair, if ;/i + y[ = 1.
Remark. - In view of Lemma 4.6 of [34] and Theorem 8.1 of [23], either ^ = 0 and y[ = 1, or z/i = 1 and v[ = 0.
LEMMA 4.12. - If{vj}^^ and {;4}^i ^^ members of a singular pair, then 7(5,^1,..., z^)7(5, ^ , . . . , z/J,
w/n'c/i w^ ^/za// ca// ^ ^/-function for the singular pair, is nonzero for Re(^) > 0.
LEMMA 4.13. - Every chain -which is neither non-negative nor non-positive is ^-equivalent to either a singular pair or a pair of regular chains starting at 1/2, with ^/-function given as the product of their ^-functions.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
Proof. - Let {^}^ be such a chain. Then
Vn > ^n-1 > . . . > ^ > 0 > Z^_i > . . . > ^i.
The pair
{ ^ , . . . , ^} U { - ^ - i , . . . , -^i}
is a pair of non-negative chains. The initial points for these are ^ and -^-i, and Vi — vi-\ = 1 implies that the pair is singular, unless ^ = —^-i = 1/2. Moreover
7(5; ^i, . . . , ^n) = 7(5; ^ . . • , ^n)7(^; -^-1, • • • , -^l).
Then {^j}^=i is 7-equivalent to the pair { ^ , . . . , ^} U { — ^ _ i , . . . , —^i}.
DEFINITION 4.14. - Let {^-}^=i and {^}^Li be a pair of o-o-chains in the ao-support of a^. We shall say {^j}^=i can be completed to {^}^Li, if m ^ n and z^ = v'^.
PROPOSITION 4.15. - Every non-negative chain can be completed to either a regular chain, a member of a singular pair, or a chain whose initial point is negative.
Proof. - Let {^-}^i be a non-negative chain. Write
7 ( 5 : ^ 1 , . . . , ^ ) =
L{s -Vn- l)L(s + ^i - 1) L(s - J^i)L(s + Vn)
Assume {vj} is not of the types mentioned in the proposition and in particular if v\ = 1, then L(s — 1)~1 does not divide L(s — 1, OQ x r)~1. By Theorem 4.4, L{s — z/i) must be cancelled by the 7-function of another chain {^}^Lr Write
, . L(s - ^
7(5;^,...,^) = L ( s - ^ - l ) L ( s ^ ^ - l )
^m) — T / , , /£(^-^)£(.+0
By Lemma 4.5, we may assume that either ^i = v'^ + 1 or v\ + v[ = 1. Suppose
^ = ^ + 1. We can then complete {^j}^=i to the chain {^j}^=i U {^}^Li. Observe that 7(5 ; ^ i , . . . , ^ ) 7 ( 5 ; ^ , . . . , ^ )
is equal to the 7-function of { ^ j } ^ i U {^}^Li. We can therefore replace the two chains with their union and continue with the argument if v[ > 0. If v\ + u[ = 1 and v[ > 0, we then have a singular pair since we may assume v\ 7^ i^[. Otherwise, we can consider {-^k}t=m ^d ^n complete {^}^=i to the chain {^j}^i U {-^k}i=m^ slnce
v\ = 1 + (-^). By Lemma 4.8, 7(5; ^ , . . . , v'^) = 7(5; -^,..., -^) and therefore again 7(5; ^ i , . . . , ^)7(5; v'^ ..., ^) is equal to the 7-function of the union. We now use the induction if —^ > 0.
PROPOSITION 4.16. - Fix o-o. Then every ao-chain is ^/-equivalent to a chain which can be completed to a chain "which is ^-equivalent to either a regular chain, a pair of regular chains, or a singular pair of chains.
Proof. - By Lemma 4.13, every chain with both positive and negative terms is 7- equivalent to either a singular pair or a pair of regular chains starting at 1/2. Moreover,
4e SfiRIE - TOME 31 - 1998 - N° 4
by Lemma 4.8, every non-positive chain can be replaced by a non-negative one. We now apply Proposition 4.15, and Lemma 4.13, if there are any mixed chains.
We are now ready to prove Theorem 4.1.
LEMMA 4.17. - The product
b
JJ L(s + (& + 1)/2 - i - 1, ao x r ) / L ( s + (& + 1)/2 - z, (TO x r)
i=l
is non-zero for Re(^) > 0.
Proof. - Clear.
We now consider contributions from cro-chains.
LEMMA 4.18. - Let {^}^=i be a regular chain with ^ = 0 or 1/2. Then
b
J j 7 ( > + ( & + l ) / 2 - z ; ^ . . . , ^ )
z=l
is non-zero for Re(.s) > 0.
Proof. - The product
^r -rj L(s + (b + 1)/2 - i - v, - l)L(s + (& + 1)/2 - i + ^ - 1)
i
111^ + (
6+
1)/
2-
%- ^W^ + (
6+
1)/
2-
z+ ^-)
can be written with a numerator (a polynomial in q~8) as
n
J] £(5 + (b + 1)/2 - 1 - ^r^ + (6 + 1)/2 - 1 + ^•)-1
j=i
n
= U £(5 + (& + 1)/2 - j - z.i)-1^ + (6 + 1)/2 - 2 + j + ^i)-1. j=i
We shall show that every factor which has a zero at s with Re{s) > 0, cancels with a factor from the denominator.
Since j > l,b > 1, and ^i ^ 0, (6 + 1)/2 - 2 + j + z^i > 0 which allows us to disregard factors
n
n ^ + ( 6 + l ) / 2 - 2 + j + ^ i ) -1. j=i
We must therefore consider those factors for which ( & + l ) / 2 - ^ - ^ i < 0 or
n > j > (&+ 1)/2 - ^i.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
Let bo be the first integer strictly greater than (6 + 1)/2 - z^i. We may assume 60 < n.
The product is then over bo < j <, n, i.e. we need to consider:
L{s + (6 + 1)/2 - 60 - ^ i ) ~1. . . L{s + (6 + 1)/2 - n - ^i)-1. The denominator is the product
n
JJ L(s +{b+ 1)/2 -b-j- ^)~lL(s + (6 + 1)/2 - & + j - 2 + ^i)-1.
3=1
For 1^1 = 0,
, _ f fc±3 6 = odd
^ o - ^ ^even,
while for v\ = 1/2
f ^ b = odd
^^ ^,^
It is easiest to consider four cases:
Case 1. - ^i = 0 and b is odd. Then bo = b^3 and the numerator of concern is L{s - I)-1... L(s - (n - (6 + 1)/2))-1.
The denominator gives
n
n £(5 - 6/2 + 1/2 - j)-1!^ - 6/2 - 3/2 + j)-1- j=i
Given an integer i, 1 < ^ < n - Hl, either -^ < Hl, in which case j = b^3 - ^ will satisfy 1 <^ j < n, using 60 ^ n, and L^-^)"1 will be cancelled by L^-^^-S^+j')"1; or b^1 < i <n — 6^1, in which case the integer j = i — b-^- satisfies l < j < n — 6 < n and L{s — i}~1 cancels off L{s — 6/2 + 1/2 — j)~1, proving the lemma in the first case.
Case 2. - v\ = 0 and 6 is even. Then 60 = b^2 and the numerator is L{s - 1/2)-1... L(s + (6 + 1)/2 - n)-1. The denominator is
n
]^[L{s - 6/2 + 1/2 -j)-1^ - b/2 - 3/2 +j)-1-
.7=1
Given 1/2 < i < n - HL1, half of an odd integer, either i <_ (6 + 1)/2, in which case the integer j = (6 + 3)/2 - i will satisfy 1 < j < (6 + 1)/2 < n, using 60 < n, and L{s - t)-1 will be cancelled by L{s - 6/2 - 3/2 + j)~1; or b^- ^ i < n - HJ-, in which case the integer j = t - (6 - 1)/2 satisfies l < j < , n - b < n and L(s - l)~1
cancels L{s - 6/2 + 1/2 - j)~1.
4° S6RIE - TOME 31 - 1998 - N° 4
Case 3. - ^i = 1/2 and b is odd. Then &o = ^ and the numerator is L(s-l/2)-l...L(s-^b/2-n)-\
while the denominator equals
n
I] L(s - b/2 - j^L^s - b/2 - 1 + j)-1. j=i
As in Case 2, given 1/2 < ^ < n - (b/2), half of an odd integer, either i ^ b/2 for which the integer j = ^ - t satisfies 1 <, j ^ (b + 1)/2 ^ n and £(^ - ^)-1 cancels L(s - b/2 - 1 + j)-1, or 6/2 < t < n - (b/2), in which case the integer j = t - (b/2) satisfies l < j < n - b < n and L(s - £)~1 cancels L(s - (b/2) - j)~1.
Case 4. - ^i = 1/2 and b is even. Then bo = (b + 2)/2 and the numerator is
L(s-l)-1 . . . £ ( 5 + & / 2 - n ) -1.
Given 1 < i < n- (b/2), an integer, either i ^ b/2 for which the integer j == (6+2)/2-^
will satisfy 1 ^ j ^ 6/2 < n and L(s -i)-1 will be cancelled by L(s - b/2 -1 +j)-1, or 6/2 < t < n - (b/2), in which case the integer j = i - (b/2) satisfies l<j<n-b<n and L(s - i}-1 is cancelled by L(s - (b/2) - j)-1.
The lemma is now complete.
LEMMA 4.19. - Let {^}7=i and {^}^i be a singular pair. Then
b
J] -r(s + (b + 1)/2 - %; ^..., ^(s + (6 + 1)/2 - ,; ^ . . . , ^) 1=1
^ non-zero for Re(s) > 0.
Pwo/. - It is enough to prove the same statement for
b
J] L(s + (b + 1)/2 - z - ^ - l ) / L ( s + (6 + 1)/2 - i + ^), i=i
as well as for
b
n ^ + ( 6 + l ) / 2 - z - ^ - l ) / ^ + ( 6 + l ) / 2 - z + ^ ) . 1=1
Suppose Vn > (b - 2)/2. Then
(b + 1)/2 - % + ^ > b - (1/2) - i > -1/2.
By Theorem 8.1 of [23] and Lemma 4.6 of [34], v^ is a half integer, since r is generic.
Consequently
( & + l ) / 2 - z + ^ ^ 0 , proving the lemma in this case.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
Now suppose 0 < Vn < (6 — 2)/2. Given an integer j, b > j > (b + 1)/2 + v^, choose an integer i, 2vn being an integer, such that i == j — 2vn — 1. Then
b > i = j - 2vn - 1 > (6 + 1)/2 - i/n - 1
> ( 6 + 1 ) / 2 - ( & - 2 ) / 2 - 1
=1/2.
This implies that i > 1 since it is an integer. Consequently the factor L(s + (b + 1)/2 — J + ^n)~1 is cancelled off by L(s + (& + 1)/2 — i — Vn — I)"1' The lemma is now proved.
LEMMA 4.20. - Let ljH}^ii
LEMMA 4.20. - Let l ) { ^ } ^ = i be the union of all the regular 0-Q-chams with y[ = 1,
^=1
1 <_ i < c, which appear in the support of a^. Then the product of
b c
^Ih( s + ( 6 + l )/ 2 - ^ ' H^i)
i=l ^=1
with b
JJ L(s + (b + 1)/2 - % - 1, ao x r ) / L { s + (& + 1)/2 - z, (TO x r) 1=1
is non-^ero for Re(s) > 0.
Proof. - Going back over the proof of Lemma 4.9, using Theorem 4.4, it follows that in fact L^s)'0 divides L^s.o-o x r)~1. Set
0(s,o-o x r ) = L(.s,o-o x TVLO)0.
Then both
b
Y[ 0{s + (b + 1)/2 - i - 1, ao x r ) / e ( s + (& + 1)/2 - z, ao x r) 1=1
(Lemma 4.17) and
c b
]^[ J ] £ ( , + ( 6 + l ) / 2 - z - < - ! ) / £ ( . + ( 6 + l ) / 2 - z + ^ )
^=1 i=i
(Proof of Lemma 4.19) are non-zero for Re(.s) > 0 and their product is equal to the product in the statement of the lemma.
Theorem 4.1 is now a consequence of Proposition 4.16, and Lemmas 4.17, 4.18, 4.19, and 4.20, applied to every regular chain or singular pair.
REMARK 4.21. - Lemmas 4.1 and 4.6 of [34] have a similar proof which is a clever application of Casselman's square integrability criterion [5]. It extends to other classical groups which are not discussed in [34] as well, when the lemmas are formulated appropriately. The only change is in the case of unitary groups which implies po,j ^ Poj' This is precisely what is needed for obtaining our results.
4'^ SfiRIE - TOME 31 - 1998 - N° 4
REMARK 4.22. - The regular chains and singular pairs defined here are the same as those defined by Tadic in [34] which is the same as in [5], and that is how we chose these terminologies. In fact our Proposition 4.16 proves some of the results of [34]. Observe that Proposition 4.16 is based on Theorem 4.4 which was proved in [23].
REMARK 4.23. - Muic [42] now also has a proof of Theorem 4.1 when G = Sp^ or SO-zn+i- His proof, although quite different, also relies on the results of [23]. The paper contains some very interesting results for these groups.
(See the introduction here.)
5. Applications
In this section we prove a result which determines the poles of intertwining operators in terms of those of L-functions whenever injectivity (Definition 3.1) holds in a certain level.
We then apply this result to determine the poles of intertwining operators in terms of Artin L-functions in an important archimedean case (Theorem 6.1, Section 6).
Let G be again a quasisplit connected reductive algebraic group over a local field F of characteristic zero as in Sections 1-3. Fix a Borel subgroup B and write B = TU, where T is a maximal torus and U denotes the unipotent radical of B.
Fix a F-parabolic subgroup P = MN with N c U and T c M, a Levi decomposition.
Let Ao, W{Ao), ^, ^M, ci*, o^, all be as in Section 1. Suppose TT is an irreducible admissible -0M-generic representation of M = M(F). Let J(^,7r), v G a^, be as in Section 1.
Assume P is maximal and let a be the unique simple root in N. As in [23], let a = (p,a)~1 ' p, where p is half the sum of roots in N. Given s € C, sd G a^. Let A(s5,7r,wo) be the standard intertwining operator from I(sa,7r) into J(wo(5a),wo(7r)), where WQ is a representative for WQ.
As in Section 1, denote by LM, the L-group of M and let Ln be the Lie algebra of
m
the L-group of N. Let r be the adjoint action of LM on Ln and decompose r = Q) r^, with ordering as in [23]. For each z, 1 < i < m, let L{s,a,ri) be the local L-function defined in [23]. (See Section 1 here.) It is defined to agree completely with Langlands definition of L-functions whenever there is a parametrization. In particular the L-function for arbitrary a is just the analytic continuation of the one attached to the tempered inducing data through the product formula (cf. part 3 of Theorem 3.5 and equation 7.10 of [23]).
(See also Theorem 5.2 of [26].)
Next, embed TT as a subrepresentation of a module IM^.CT) = IndM^fM^ ^ 1,
where a is in the discrete series and v is in the closure of the negative Weyl chamber of Oe (Langlands classification). Here dg is the real Lie algebra of the split component Ae of the center of Me.
As usual fix s G C. Embed I(sa, TI-) ^ I{sa + v, a), where sa in I(sa + v, a) denotes an extension of sa to a^. Let WQ denote the longest element in the Weyl group of Ao in G modulo that of Ao in M. Fix a reduced decomposition WQ = Wn-i... wi with respect
ANNALES SCIENTIFIQUES DE L'^COLE NORMALE SUPERIEURE
to the Levi subgroup M^ of G (Lemma 2.1.1 of [25]). For each j, 2 < j < n — 1, there exists a unique simple root aj such that Wj^aj) < 0. Let wj = W y - i . . .wi with wi == 1. Let f^ = 0j U {c^}, where 0 ^ = 0 and 6^-+i = Wj(0j), 1 < j ^ n - 1. The group MQ^ contains M^ as the Levi subgroup of a maximal parabolic subgroup. For each j\ ay = Wj{a^) is an unramified twist of a discrete series of M^. Let J^(ay) denote the corresponding induced representation of M^. Write J^(a^) = J(^,cr^), where cr^ is in the discrete series. Up to an unramified twist a^ is unique. We will assume that for each j, a'j is such that every standard module of M^ which has a'y as its tempered inducing data satisfies injectivity (Definition 3.1).
THEOREM 5.1. - Suppose Conjecture 1.1 is valid -whenever F is p-adic, e.g. G is classical.
Moreover assume for each j every standard module of MQ^ which has a'y as its tempered inducing data satisfies injectivity (Definition 3.1). Then
m
JJ L{is, TT, r^A^sa, TT, wo) 1=1
is entire.
Proof. - The intertwining operator A(5a,7r,wo) is a restriction of the product of rank one operators A{^j^a^Wj)^ 1 <: j <: n — 1. So are the L-functions L(5,7r,r^), of course under validity of Conjecture 1.1 if F is non-archimedean (Part 3 of Theorem 3.5 and equation (7.10) of [23], as well as Theorem 5.2 of [26]). One must therefore prove the following lemma.
THEOREM 5.2. - Theorem 5.1 is valid if7r is in the discrete series.
Proof. - We need to show that if I{sa^7r) satisfies injectivity for all s G C, with Re(^) > 0, then the theorem is valid.
m
If Re(^) > 0, then A(5Q,7r,wo) and ]~[ £(%5,7r,r^) are both holomorphic. For s with
_ i=l _
Re(^) = 0 which is a pole of A(5a,7r,wo), I{sa^) is irreducible by the theory of Ji-groups. Consequently the local coefficient C^^sa^ir) must have a zero of the same
m
multiplicity (cf. equation (1.2) of [23] and Section 1 here). Since ]~[ £(1 - %5,7r,r^)
m 1=1
is holomorphic for Re(.s) = 0, the same is true about f] L(is^7r^ri)~1. It remains to consider the case of Re(5) < 0. 1=1
Suppose Re(^) > 0. Given / in an irreducible subspace of I{sa^7r), there exists a ^ extending ^M such that \^(sa,7r){f) / 0 by injectivity assumption. Thus it follows from
(5.1) A^(5a, TI-) = C^{sa, 7r)\^{wo{sa), wo(7r))A(5a, TT, wo)
and equation (3.11) of Theorem 3.5 of [23] that C^{sa,7r)A{sa,7r^wo) must never be zero for Re(^) > 0. Using
C'^(wo(5a),wo(7^))A(wo(<55),wo(7^),Wo"l)C7^(55,7^)A(55,7^,wo) = I it is now clear that
C^{wo(sa), wo{7r))A(wo{sa), wo(7r), w^1))
4'^ SfiRIE - TOME 31 - 1998 - N° 4
is holomorphic for Re(^) > 0. This implies that
m
JJ L(l - is, TT, ri)/L{is, TT, r,) • A(5a, TT, wo) 1=1
is holomorphic whenever Re(.s) < 0, completing the lemma.
With respect to reducibility of standard modules one has the following
PROPOSITION 5.3. - Let F be any local field. But in the case of p-adic F, assume that G satisfies Conjecture 1.1. Let P = MN be a maximal parabolic subgroup of G and fix a generic irreducible tempered representation a- of M. Suppose Re(s) > 0. Assume Vegan's theorem is valid for every I(sa,a), i.e. if J(sa,a) is generic, then I{sa,a) is
m
irreducible. Then I(sa, a) is irreducible if and only if ]~[ £(1 - is, a, r^)"1 ^ 0. When F is
i=l
archimedean, the L-functions are those ofArtin attached by Langlands and the assumption is already a theorem (Theorem 6.1 of [37]).
Proof. - If I{sa, a) is reducible, then by our assumption (Vogan's Theorem 6.1 of [37]
if F is archimedean) its Langlands quotient is not generic. Consequently
\^{wo(sa), Wo(a))A(sa, a, wo) will be identically zero. But X^(sa^a) is not zero. Consequently
m
C^{sa,a,WQ)~1 = 0 which implies f] ^l - is,a,ri)~1 = 0 (equations (3.11) and (7.4)
m i=l
of [23]) since ]~[ L{is,a,ri)~1 is non-zero for Re(<?) > 0.
i=l
Conversely suppose I{sa,a) is irreducible. Then \^(wo(sa),wo{a))A{sa,a,wo) is
m
never zero and therefore n -H1 - ^,cr,r,)~1 -^ 0 since \^(sa,a) is holomorphic.
1=1
More generally we have the following conditional reducibility criteria for representations induced from irreducible generic quasi-tempered representations. Applying standard arguments, such as inducing in stages for singular parameters, we may assume that their complex parameters are in the positive Weyl chamber.
PROPOSITION 5.4. - Let F be any local field. Suppose Vegan's theorem is valid for the standard modules of G. Let P = MN be an arbitrary parabolic subgroup of G, P D B, and fix an irreducible \-generic tempered representation ofM. Let v G a^ be in the positive Weyl chamber of the split component ofM.. Let C^(^a) be the local coefficient attached to v^a, and '0. Then J(^,a) is irreducible if and only if C^^a)"1 7^ 0.
Proof. - Exactly as in Proposition 5.3.
6. An Important Archimedean Case
In this section we will apply Theorem 5.1 to an important special case when F is archimedean. The case in hand has an important application in lifting of automorphic forms from classical groups to GLr as being pursued in [11,30,31], using the converse theorem [9].
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
More precisely, let F = R or C and let G = SO^ the split special orthogonal group of rank n. We will be concerned only with its Siegel parabolic subgroup P = MN for which M ^ GLn. Let a be an irreducible admissible generic representation of M = GLn(F).
Let WF be the Well group of F/F (cf. [1,17,18,24]) and fix a representation (f):WF-^LM= GLn(C)
which parametrizes a as in [17]. Given a representation r of LM = GL^(C), let L(s, r • (f)}
be the Artin L-function attached to the representation r • (f) of Wp. (See [1, 18, 24].) Finally, note that the adjoint action of LM = GLn(C) on Ln, the Lie algebra of the L-group of N, is equal to A2^, the exterior square representation of GLn{C) (cf. [27]). We shall prove:
THEOREM 6.1. - Let G = SO-zn and F = R or C. Assume P = MN is the Siegel parabolic subgroup of G. Let a be an irreducible admissible generic representation of M.
Choose the homomorphism (representation)
(j): WF -^ GLn(C)
parametrizing a. Let A{sa. cr.wo) be the standard intertwining operator discussed before.
Then as a function of s
L(s, A2?^ ' (j^^A^sa, a, Wo) is entire, where (f) is the contragredient of (f).
Proof. - By Vogan's results (Theorem 6.2.f of [37]) and the fact that 7?-groups for GLn are trivial, one concludes that a is in fact a full induced representation, induced from a tensor product of essentially discrete series representations of a product ]~[^ GL^(F).
Since F = R or C, m, == 1 or 2.
Going back to Theorem 5.1, we only need to prove that the inject! vity holds in each of the rank one cases. Then G is either GLmW, m = 2,3,4, GL^(C), or finally split 504(R). The Levi subgroups for m = 3 and 4 are GL^ x GL^ and GL^ x GL^, respectively, while that of SO^ is the Levi subgroup of the Siegel parabolic subgroup, i.e. M ^ GL^. The L-functions in [23] are now precisely those of Artin mentioned above (Theorem 3.5 of [23]).
We shall now check the injectivity in each of the above cases. More precisely, we must show that in each case, every representation induced from an essentially discrete series data whose central character is in the positive Weyl chamber, i.e. a standard representation, contains no non-generic irreducible subspaces.
When G = GL^ this is well known. Suppose G = 504 (R). Realize the (topological) connected component of G as the quotient of SL^(R) x SL'z(R) by {±1}. The (topological) connected component of M = GL^(R) is the image of SL^{R) x R* with R* realized as the diagonal subgroup of the second SL^(R) in this product. The induction corresponds to a principal series in 5'£2(R). The injectivity is then a consequence of the same fact for SL^ (R).
It remains to consider GL^W with m = 3 or 4. The result must be contained somewhere in Speh's thesis [32]. In fact m = 3 is clearly there. But, one expects it to be valid for any standard module of GL^(R) for arbitrary m, and this is in fact the case and a proof of it
4e S6RIE - TOME 31 - 1998 - N° 4
